Exponential growth
The quantity grows at a rate directly proportional to its present size.
If the constant of proportionality is negative, then the quantity decreases over time, and is said to be undergoing exponential decay instead.
Growth like this is observed in real-life activity or phenomena, such as the spread of virus infection, the growth of debt due to compound interest, and the spread of viral videos.
In real cases, initial exponential growth often does not last forever, instead slowing down eventually due to upper limits caused by external factors and turning into logistic growth.
After one hour, or six ten-minute intervals, there would be sixty-four bacteria.
Many pairs (b, τ) of a dimensionless non-negative number b and an amount of time τ (a physical quantity which can be expressed as the product of a number of units and a unit of time) represent the same growth rate, with τ proportional to log b.
For any fixed b not equal to 1 (e.g. e or 2), the growth rate is given by the non-zero time τ.
For any non-zero time τ the growth rate is given by the dimensionless positive number b.
Thus the law of exponential growth can be written in different but mathematically equivalent forms, by using a different base.
Parameters (negative in the case of exponential decay): The quantities k, τ, and T, and for a given p also r, have a one-to-one connection given by the following equation (which can be derived by taking the natural logarithm of the above):
A popular approximated method for calculating the doubling time from the growth rate is the rule of 70, that is,
, then the log (to any base) of x grows linearly over time, as can be seen by taking logarithms of both sides of the exponential growth equation:
For example, if one wishes to empirically estimate the growth rate from intertemporal data on x, one can linearly regress log x on t. The exponential function
In the above differential equation, if k < 0, then the quantity experiences exponential decay.
For a nonlinear variation of this growth model see logistic function.
There is a whole hierarchy of conceivable growth rates that are slower than exponential and faster than linear (in the long run).
See Degree of a polynomial § Computed from the function values.
Growth rates may also be faster than exponential.
In reality, initial exponential growth is often not sustained forever.
After some period, it will be slowed by external or environmental factors.
[10] Exponential growth models of physical phenomena only apply within limited regions, as unbounded growth is not physically realistic.
Although growth may initially be exponential, the modelled phenomena will eventually enter a region in which previously ignored negative feedback factors become significant (leading to a logistic growth model) or other underlying assumptions of the exponential growth model, such as continuity or instantaneous feedback, break down.
Studies show that human beings have difficulty understanding exponential growth.
[11] According to legend, vizier Sissa Ben Dahir presented an Indian King Sharim with a beautiful handmade chessboard.
The king readily agreed and asked for the rice to be brought.
trillion) on the 41st and there simply was not enough rice in the whole world for the final squares.
(From Swirski, 2006)[12] The "second half of the chessboard" refers to the time when an exponentially growing influence is having a significant economic impact on an organization's overall business strategy.
French children are offered a riddle, which appears to be an aspect of exponential growth: "the apparent suddenness with which an exponentially growing quantity approaches a fixed limit".
The riddle imagines a water lily plant growing in a pond.
Day after day, the plant's growth is small, so it is decided that it won't be a concern until it covers half of the pond.
